Video Transcript
In this video, we will learn how to
find the coordinates of a midpoint between two points or those of an endpoint on the
coordinate plane. Before we look at a coordinate
plane, let’s look at a midpoint on a number line.
If we start with a number line and
we have points at two and eight, where is the midpoint between two and eight? You might say, “Well, it’s going to
be halfway between two and eight.” Someone else might say, “It’s the
average of two and eight.” The average of two and eight is
halfway between. To find the average of two and
eight, we would add two plus eight and then divide by two, which equals 10 divided
by two, which is five. And that means halfway between two
and eight, the average of two and eight is five.
To find the midpoint on a
coordinate plane, we use a really similar process. Let’s consider this line segment in
a coordinate grid. If we label the blue point 𝐴, 𝐴
is located at two, two. And if we make the other endpoint
𝐵, 𝐵 is located at eight, six. And 𝐶 is the midpoint. This means that the distance from
𝐴 to 𝐶 is equal to the distance from 𝐶 to 𝐵. It also means that 𝐶 is halfway
between 𝐴 and 𝐵 in the vertical direction and 𝐶 is halfway between 𝐴 and 𝐵 in
the horizontal direction. And we write that mathematically
like this.
To find the 𝑥-coordinate of the
midpoint, we take the 𝑥-coordinates of the two endpoints and we average them. Remember that the endpoints are
located at either end of the line segment. Then, the 𝑦-coordinate of the
midpoint is found by taking the 𝑦-coordinates of the two endpoints and averaging
those. In this case, we can let 𝐴 be 𝑥
one, 𝑦 one and 𝐵 be 𝑥 two, 𝑦 two. If we want to locate point 𝐶’s
coordinates, we plug in the values we know from the endpoints. 𝑥 one is two. 𝑥 two is eight. 𝑦 one is two. 𝑦 two is six.
When we add those together, we get
10 over two, eight over two, which simplifies further to five, four. The midpoint is then located at the
point five, four. Five is halfway between two and
eight and four is halfway between two and six. Using this formula, we can find the
midpoint if we’re given two coordinates. Or we can find the coordinates of
an endpoint if we’re given one endpoint and the midpoint.
Let’s look at some examples.
On the graph, which point is
halfway between one, eight and five, two?
We have the point one, eight — we
could call it 𝐴 — and the point five, two — we can call 𝐵. And we want to know which point is
halfway between the two points. The point halfway between these two
points is called the midpoint. And to find the midpoint, we
average the 𝑥-coordinates of the two endpoints. And then, we average the
𝑦-coordinates of the two endpoints. So, we’ll let point 𝐴 be 𝑥 one,
𝑦 one and point 𝐵 be 𝑥 two, 𝑦 two. Then to find the midpoint, we add
one plus five and divide by two. And then, we add eight plus two and
divide by two. This gives us six over two and 10
over two, which we reduce to three, five.
On the graph, the coordinate three,
five is here. And if we look at this point, we
see from 𝐵 that is up three and left two. And then, if we want to go from the
midpoint to point 𝐴, again, we have up three and left two. This confirms that the distance
from 𝐴 to the midpoint is the same as the distance from the midpoint to 𝐵. Halfway between one, eight and
five, two is the point three, five.
Here’s another example. This time, we know the
endpoint. But we’re missing part of the
coordinates for the endpoints.
Consider the points 𝐴: 𝑥, seven;
𝐵: negative four, 𝑦; and 𝐶: two, five. Given that 𝐶 is the midpoint of
line segment 𝐴𝐵, find the values of 𝑥 and 𝑦.
First, let’s list out what we
know. We have line segment 𝐴𝐵 and 𝐶 is
the midpoint point. Point 𝐴 is located at 𝑥,
seven. Point 𝐵 is located at negative
four, 𝑦. And point 𝐶 is located at two,
five. At this point, you might be
thinking, “Shouldn’t we try to graph these values?” But because we’re missing this 𝑥-
and 𝑦-value, it’s not easy to graph this. So, let’s consider what we know
about the midpoint.
The coordinates of the midpoint can
be found by averaging the 𝑥-coordinates of the endpoints and the 𝑦-coordinates of
the endpoints. And if the midpoint is two, five,
then 𝑥 one plus 𝑥 two divided by two has to be equal to two. And 𝑦 one plus 𝑦 two divided by
two has to be equal to five. So, we set up two separate
equations, one that says 𝑥 one plus 𝑥 two over two equals two and one that says
five equals 𝑦 one plus 𝑦 two over two. We’ll let 𝐴 be 𝑥 one, 𝑦 one and
𝐵 be 𝑥 two, 𝑦 two and then we plug in what we know.
Two is then equal to 𝑥 plus
negative four divided by two and five is equal to seven plus 𝑦 divided by two. And now, we just need to solve for
each variable. On the left, we multiply both sides
of the equation by two, which will give us four equals 𝑥 plus negative four, which
we can rewrite to say four equals 𝑥 minus four. Then add four to both sides. And we see that eight equals 𝑥 or,
more commonly, 𝑥 equals eight. We follow the same procedure to
solve for 𝑦. Multiply by two. Subtract seven from both sides. Three equals 𝑦. So, 𝑦 equals three. Since 𝐴 equals 𝑥 seven and 𝑥
equals eight, point 𝐴 is located at eight, seven. And since 𝐵 was located at
negative four, 𝑦 and 𝑦 equals three, 𝐵 is located at negative four, three.
Here’s another example, where we
already know the endpoint and we need to find the coordinates of the endpoints.
Find the point 𝐴 on the 𝑥-axis
and the point 𝐵 on the 𝑦-axis such that three over two, negative five over two is
the midpoint of line segment 𝐴𝐵.
First, let’s list out what we
know. 𝐴 is located along the
𝑥-axis. 𝐵 is located along the
𝑦-axis. And we have a midpoint of
three-halves, negative five-halves. At this point, we should think
about what we know about the 𝑥- and 𝑦-axis. Values along the 𝑥-axis have a
𝑦-coordinate of zero. And that means if point 𝐴 is
located along the 𝑥-axis, we don’t know where along the 𝑥-axis that is. But we do know it’s zero along the
𝑦-axis. We can write out the coordinates of
point 𝐴 as 𝑎, zero. And in a similar way, points along
the 𝑦-axis have zero as their 𝑥-coordinate. That means we can write point 𝐵 as
being located at zero, 𝑏. And finally, we remember that we
can find the midpoint by averaging the 𝑥-coordinates of the endpoints and the
𝑦-coordinates of the endpoints.
Using this information, we can set
up two equations to solve for our missing values. We’ll let point 𝐴 be 𝑥 one, 𝑦
one and point 𝐵 be 𝑥 two, 𝑦 two. We plug that information into our
midpoint formula. That means three-halves is equal to
𝑎 plus zero over two and negative five-halves equals zero plus 𝑏 over two. 𝑎 plus zero equals 𝑎. Three-halves must be equal to 𝑎
over two. And in the same way, negative five
over two is equal to 𝑏 over two, which means 𝑎 equals three and 𝑏 equals negative
five. We take this information and plug
it back into our points. And we get the point three, zero
and the point zero, negative five. These two points have the given
midpoint.
Let’s look at a case where we can
use what we know about midpoints to solve a problem.
A rectangular garden is next to a
house along the road. In the garden is an orange tree
seven meters from the house and three meters from the road. There is also an apple tree, five
meters from the house and nine meters from the road. A fountain is placed halfway
between the trees. How far is the fountain from the
house and the road?
We have a garden that is next to a
house along a road. In the garden, there are two trees,
an orange tree that’s located seven meters from the house and three meters from the
road. And then, there’s an apple tree,
five meters from the house and nine meters from the road. A fountain is placed halfway
between them. How can we find out how far the
fountain is from the house and from the road?
We could write these distances in a
coordinate form, where the 𝑥-coordinate is the distance from the house and the
𝑦-coordinate is the distance from the road. The orange tree is then located at
seven, three. The apple tree is located at five,
nine. And the fountain is the
midpoint. And we know to find the midpoint,
we add the 𝑥-coordinates of the endpoints and divide by two and then add the
𝑦-coordinates of the endpoints and divide by two. Which means the fountain is located
at seven plus five divided by two and nine plus three divided by two, which is 12
over two, 12 over two or six, six. We know that our 𝑥-coordinate is
the distance from the house and the 𝑦-coordinate is the distance from the road. Therefore, the fountain is located
six meters from the house and six meters from the road.
In our last example, we’ll look at
a midpoint that is the midpoint of two different line segments at the same time.
Suppose 𝐴 is negative seven,
negative four; 𝐵 is six, negative nine; and 𝐷 is eight, negative two. If 𝐶 is the midpoint of both line
segment 𝐴𝐵 and line segment 𝐷𝐸, find point 𝐸.
Let’s first sketch what we
know. We have a line segment 𝐴𝐵 with
the midpoint 𝐶. And 𝐶 is also the midpoint of line
segment 𝐷𝐸. We’re given the coordinates for 𝐴,
𝐵, and 𝐷. Our end goal is to find the
coordinates of point 𝐸. But before we can find 𝐸, we’ll
need to know 𝐶. Once we find 𝐶, we can find
𝐸. And to do both of these things,
we’ll need to remember that the midpoint formula looks like this.
The 𝑥-coordinate of the midpoint
is found by taking the 𝑥-coordinates from the endpoints and dividing by two. And the 𝑦-coordinate of the
midpoint is found by averaging the 𝑦-coordinates of the two endpoints. Since 𝐶 is the midpoint of 𝐴 and
𝐵, we’ll let 𝐴 be 𝑥 one, 𝑦 one and 𝐵 be 𝑥 two, 𝑦 two. The midpoint 𝐶 will be located at
negative seven plus six over two, negative four plus negative nine over two. Negative seven plus six over two is
negative one-half. And negative four plus negative
nine is negative 13. So, the 𝑦-coordinate is negative
13 over two. Now, we know where 𝐶 is
located. And we’re ready to think about
𝐸.
If 𝐶 is also the midpoint of 𝐷𝐸,
then the coordinates of 𝐶 will be equal to the 𝑥-coordinates of 𝐷 and 𝐸 averaged
together and the 𝑦-coordinates of 𝐷 and 𝐸 averaged together. We’re given the coordinates of
𝐷. That’s eight, negative two. And so, we plug that in. From here, we’ll make two separate
equations. We’ll set negative one-half equal
to eight plus the 𝑥-coordinate of 𝐸 over two. And negative 13 over two is equal
to negative two plus the 𝑦-coordinate of 𝐸 over two. We’ll give ourselves a little bit
more room.
Since all of the denominators are
two, then the numerators are equal to each other. Negative one equals eight plus the
𝑥-coordinate of 𝐸. And to solve for that missing
value, we subtract eight from both sides. And we see the 𝑥-coordinate for
point 𝐸 is negative nine. To solve for the 𝑦-coordinate of
point 𝐸, we add two to both sides. The 𝑦-coordinate of 𝐸 is negative
11. In coordinate form, point 𝐸 is
located at negative nine, negative 11.
Each of these examples has shown us
that when you have the endpoints 𝑥 one, 𝑦 one; 𝑥 two, 𝑦 two, the midpoint is
found by adding 𝑥 one plus 𝑥 two and dividing by two then 𝑦 one plus 𝑦 two
divided by two. We can also use this formula to
find an endpoint if we know one endpoint and the midpoint. Using this formula, you’re ready to
try some on your own.
